Theorem lt0neg1

Description: Comparison of a number and its negative to zero. Theorem I.23 of Apostol p. 20. (Contributed by NM, 14-May-1999)

	Ref	Expression
Assertion	lt0neg1	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 < 0 ↔ 0 < - 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		0re	⊢ 0 ∈ ℝ
2		ltneg	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ∈ ℝ ) → ( 𝐴 < 0 ↔ - 0 < - 𝐴 ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 < 0 ↔ - 0 < - 𝐴 ) )
4		neg0	⊢ - 0 = 0
5	4	breq1i	⊢ ( - 0 < - 𝐴 ↔ 0 < - 𝐴 )
6	3 5	bitrdi	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 < 0 ↔ 0 < - 𝐴 ) )